A translation of N.M. Katz's "Exponential Sums"

TL;DR

This paper explores the theory of exponential sums over finite fields, their associated L-functions, and cohomological interpretations, providing both expository background and new results on their uniformity and specific structures.

Contribution

It offers new theorems on the cohomological uniformity of exponential sums for almost all primes and detailed analysis of particular classes with attractive algebraic geometry.

Findings

Uniformity theorems for exponential sums for almost all primes

Cohomological structure analysis of specific exponential sum classes

Connections between exponential sums and L-functions via cohomology

Abstract

These notes are devoted to the theory of exponential sums over finite fields. The first chapter recalls some of the number-theoretic interest of such sums. The second chapter discusses the $L$-functions attached to such sums, the "Weil conjectures" for these $L$-functions as established by Deligne, andthe consequences for the exponential sums themselves. The third chapter is devoted to the cohomological interpretation of exponential sums and of their associated $L$-functions. These first three chapters are largely of an expository nature. The main results are found in chapters four and five. Chapter four is devoted to theorems of uniformity "for almost all $p$" for the cohomological structure of quite general exponential sums. Chapter five is devoted to a precise analysis of the cohomological structure of certain specific classes of exponential sums for which the associated algebro-geometric situation is especially attractive.